Integration by partial fractions

[Spectre32]

Ok this is the Integral:

(x^2-1)/((x+2)^2(x+3))


Now What i did is break this up into the A + B+C ...etc etc and i came to this:


A/(x+2)^2 + Bx+C/(x+2) + D/X+3... Now i know i got to use systems of equations but I've been working on this for like 40 mins and i still can't get it straight. If anyone can help me power through this i should then be set.

 

[Parth Dave]

multiply everything by (x+2)^2(x+3). After that if you set x=-3 than everything will cancel out except D, which you can than solve for. If you set x=-3 you can create an equation that you can solve for later.

Here comes the annoying part. Expand everything out. Than group everything that has the same degree polynomial (ie group everything that is x^2, x^1, x^0) and factor out the x term. You'll notice that all of this must be equal to x^2-1. Thus, whatever you have infront of x^2 must be 1. Whatever you have infront of x must be 0. Everything else would have to be -1. You have a bunch of equations and 3 more variables to solve for. Enjoy.

i got A=0, D=8, C=-7, B=3.

 

[Hurkyl]

(Note to Spectre: you need to be more careful about using parentheses)

Why do you have (Bx + C) / (x + 2)?

 

[Parth Dave]

Its supposed to be (Ax + B) / (x + 2)^2 + C/(x + 2) + D/(x + 3)...

 

[Hurkyl]

Why do you have (Ax + B) / (x + 2)^2?

 

[Spectre32]

Yeah I know... i broke it up and then everything went to plan.. thanks.